Prove that in a Boole ring with elements there exists s.t. for all distinct .
since the ring $\displaystyle R$ is Boolean, it is unitary, commutative and has no non-zero nilpotent element. hence $\displaystyle R$ is the direct sum of $\displaystyle n$ copies of $\displaystyle \mathbb{F}_2.$ now for any $\displaystyle 1 \leq i \leq n$ let $\displaystyle a_i=(x_{i1}, \cdots , x_{in}) \in R,$
where $\displaystyle x_{ii}=1$ and $\displaystyle x_{ik}=0, \ \forall k \neq i.$ obviously $\displaystyle a_ia_j=0, \ \forall i \neq j. \ \ \Box$